Coalescence of Euclidean geodesics on the Poisson-Delaunay triangulation

Coupier, David; Hirsch, Christian (2018): Coalescence of Euclidean geodesics on the Poisson-Delaunay triangulation. In: Bernoulli, Vol. 24, No. 4A: pp. 2721-2751

Abstract

Let us consider Euclidean first-passage percolation on the Poisson-Delaunay triangulation. We prove almost sure coalescence of any two semi-infinite geodesics with the same asymptotic direction. The proof is based on an argument of Burton-Keane type and makes use of the concentration property for shortest-path lengths in the considered graphs. Moreover, by considering the specific example of the relative neighborhood graph, we illustrate that our approach extends to further well-known graphs in computational geometry. As an application, we show that the expected number of semi-infinite geodesics starting at a given vertex and leaving a disk of a certain radius grows at most sublinearly in the radius.

• Item Type: Journal article
• Faculties: Mathematics, Computer Science and Statistics > Mathematics
• Subjects: 500 Science > 510 Mathematics
• ISSN: 1350-7265
• Language: English